5. Phonons Thermal Properties
• Phonon Heat Capacity
• Anharmonic Crystal Interactions
• Thermal Conductivity
Phonon Heat Capacity
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Planck Distribution
Normal Mode Enumeration
Density of States in One Dimension
Density of States in Three Dimensions
Debye Model for Density of States
Debye T3 Law
Einstein Model of the Density of States
General Result for D(ω)
dQ  dU  PdV   U  dT   U  dV  PdV
 T V
 V T
 U   U 
  V  
 U 
  V 
 


P
dT


P
 


 



 dP



 T V  V T
  T  P 
 V T
  P T
 Q   U 
CV  
 


T

V  T V
 Q 
  V 
 U   U 
CP  


P
 
  T 
 T  P  T V  V T
P

 S   V 
 CV  T 
 

 V T  T  P
2
 P   V 

P

V




 CV  T 
 
  CV  T 
 

 T V  T  P
 V T  T  P
where
P 
1  V 


V  T  P
 P2
 CV  TV
T
= Thermal expansivity
1  V 
T   
 = isothermal compressibility = 1/ B
V  P T
CP  CV  BTV  P2
 9BTV  2
B = Bulk modulus
α = linear (1-D) thermal expansivity
Lattice heat capacity:
 U lat 
Clat  


T

V
U lat   U K , p   nK , p
K
p
K
Planck distribution:
n 
p
1
e
 / k BT
1
K , p
p = polarization
Planck Distribution
P  E   e  E / kB T
System at constant T  Canonical ensemble :
Boltzmann factor
N n 1
 E E /k T
 e  n1 n  B  e 
Nn
For a set of identical harmonic oscillators
 / kB T
Nn = number of oscillators in the nth excited state when system is in thermal equilibrium
Pn 
Probability of an oscillator in the nth excited state:
Nn
N
s 0

Occupation number:

n   nPn
n 0

 ne
n0

e
 n  / kB T
 s  / kB T
s 0

1
x


1 x
s 0
s
x  e
 / kB T

x
d  s

sx

x
x
2


dx
1

x


s 0
s 0
s
n 
x

1 x
e
1
 / k BT
1


s
e n

e
s 0
 / kB T
 s  / kB T
Normal Mode Enumeration
U lat  
K
p
K , p
e
 K , p / k BT
Clat    d Dp  
p
1

   d  D p  
p
  e /k T
k BT 2  e
2
B
 / k BT
2
 1
2
 
e  / k BT
 k B   d  D p   

2
k
T
p
 B   e  / kBT  1

e
 / k BT
1
Density of States in One Dimension
Fixed boundary problem of N+1 particles.
N = 10
L  Na
u0  t   uN  t   0
→ u  K , p, t   u  p, 0  eiK , p t sin sKa
s
 u  p, 0  e
 i K , p t
sin
with
K m
sm
N

L

m
Na
m  0,1,2, , N
Number of allowed K for non-stationary solutions is N–1 = Number of mobile atoms
Polarization p : 1 long, 2 trans
DK  
L

 /a


/ Na
dK D  K   N  1
Periodic boundary problem of N particles
L  Na
N=8
us  t   uN  s  t 
→
with
u s  K , p , t   u  p, 0  e
K  2m

L

 i K , p t i s K a
2m 
Na
us  K , p, t   u  p, 0  e
e
K 

a
 i K , p t i s 2 m / N
e
→
m  0, 1, , 
N 1 N
,
2 2



 K   are the same 
a


Number of allowed K for non-stationary solutions is N = Number of mobile atoms
L
DK  
2
 /a


 /a
dK D  K   N
fixed B.C.
 /a
max
0
0
 D  K  dK  
DK  
dK
DK 
d 
d
L
→

max
 D   d
0
D    D  K 
dK
L 1

d
 vG
Periodic B.C.
max
 /a
dK
D
K
dK

D
K
d 






d

 / a
max
DK  
L
2
→
max
 D   d
0
D    2 D  K 
dK
L 1

d
 vG
Density of States in Three Dimensions
Periodic B.C. ; N 3 cells in cube of side L → density of states in K-space is
3
V
 L 
D K   


3
 2  8
Number of modes per polarization lying between ω and ω + d ω is
D   d 

d K D K 
3
    K    d
→ density of states in ω-space is
For isotropic materials,
D   
V
 3
8

dK d 2 S
K   
V
d 2S
 3 d 
8
 K
K   
V
d 2S
D    3 
8   K    K
 K     K 
V
2 dK
4

K
8 3
d
→
 K 
V
d 2S
 3 
8   K    vg
d
dK
Debye Model for Density of States
Debye model:
  K   vK
v  velocity of sound (for a given type of polarization)
V K2
 2
2 v
V
dK
D    3 4 K 2
8
d
V 2
 2 3
2 v
For a crystal of N primitive cells:
N
D
 D   d 
0

D
V
2 v
2 3
 d 
0
2
V D3
 2 3
6 v
1/3
→
N

D   6 2 v 3 
V 

D
 Debye frequency
1/3
N

KD 
  6 2 
v 
V 
Thermal (vibrational) energy 
U
D
 d  D   n  

0
U
where
D
V
2 v
2 3
x
 d 

2
e
0
 / kBT

V kB4T 4
 2 3 3
1
2 v
xD 
k BT
D
k BT
θ  Debye temperature 
T 
U  3Nk BT  
 
V
CV  2 3
2 v
D

0
3 xD
x3
0 dx e x  1
 dx
0
Debye integrals:
See Ex on Zeta
functions, Arfken

T
D
kB
x3
0 dx ex 1
1/3
v  6 2 N 
 

kB  V 
3 xD
3
 
d

T  e
T 
 3NkB  
 

xD
 / k BT
V



2 2 v3
1 
x 4e x
e
x
 1
for each acoustic branch
2
D

0
d
e
3
 / k BT

 1 k BT
2
2
e
 / k BT
Debye
model
1 cal/mol-K  4.186 J/mol-K
Debye T3 Law

For low T, xD →  :
→
T 
U  3Nk BT  
 



x3
4
4
3
s x
 6  s  6   4 
e
0 dx e x  1  0 dx x 
15
s 1
s 1
3 xD
x3
0 dx e x  1
4 4
T 
CV 
Nk B  
5
 
3
4
T 

Nk BT  
5
 
T 
78 Nk B  
 
so that
for each acoustic branch
3
To account for all 3 acoustic branches, we set
and
3
1/3
v  6 2 N 
 

kB  V 
V 2
D     2 3
vp
p 2
1  1 2   kB   V 

 3  3    2 
3

3  vL vT     6 N 
3
1
12 4
T 
CV 
Nk B  
5
 
3
T 
234 Nk B  
 
3
Good for T < θ /50
Solid Ar, θ = 92K
c.f. Table for Bulk Moduli, Chap 3, p.52
Qualitative Explanation of the T3 Law
Of the 3N modes, only a fraction (KT /KD ) 3 = ( T / θ )3 is excited.
U
T 
3 Nk BT  
 
3
→
T 
C 12 Nk B  
 
3
Einstein Model of the Density of States
N oscillators of freq ω0.
U
D

d  D   n  
D    N    0 
  N n 0 
0
0 
N 0
e 0 / k B T  1
2
 0 
U 
e 0 / kBT
CV  

  Nk B 

T
k
T

V
 B  e 0 / kBT  1


2
Diamond
E 
0
kB
 1320 K
Classical statistical mechanics:
Dulong-Petit value CV = 3NkB
General Result for D(ω)
V
d 2S
D    3 
8   K    vg
Si
Debye solid
vg ~ 0  Van Hove Singularities
Anharmonic Crystal Interactions
Harmonic (Linear) Waves:
• Normal modes do not decay.
• Normal modes do not interact.
• No thermal expansion.
• Adiabatic & isothermal elastic constants are equal.
• Elastic constants are independent of P and T.
• C → constant for T > θ .
Deviation from harmonic behavior → Anharmonic effects
Thermal Expansion
U  cx 2  gx3  fx 4
1-D anharmonic potential:

x 
High T:
 dx x e


 dx e
 U
Boltzmann
distribution
 U

1
k BT

g, f


dx x n e a x 
1
2



4
dx x e
a x
2

g
x 
e  c x 1   g x3   f x4 
e  U
→
c
2 a n 1
2
 n 1 
i n
1

e


  2 
3
4

5
  c

c


even
 n  1!!
n n 1

for n  
2 a
 odd

0


3 

4 a5

dx e



3g
3g

kT
4 c 2 4c 2 B
 a x2


a
Thermal Expansion
c=1
g = .2
f = .05
Lattice constant of solid argon
Thermal Conductivity
Heat current density:
J Q   T
κ = Thermal conductivity coefficient
For phonons,
JQ = JU .
Key features of kinetic theory (see L.E.Reichl, “A Modern Course of Statistical Mechanics”, §13.4 ):
1.Quantities not conserved in particle collisions are quickly thermalized to (global)
equilibrium values. (e.g., velocity directions & magnitudes )
2.Conserved quantities can remain out of global equilibrium (e.g., stay in local
equilibrium. They get transported spatially in the presence of a “gradient”.
3.MFP is determined by collisions that do not conserve the total momenta of particles.
Net amount of A(z) transported across the x-y plane at z0 in the +z direction per
unit area per unit time:
n v  A  z0  z   A  z0  z  
2n v z
dA
dz
 2n v al
dA
dz
Δz = distance above/below plane at which particle suffered last collision.
n = particle density, l = mean free path, a = some constant.
J A  z   bA n v l
dA
dz
1
J A   n v l A
3
bA = 1/3 is determined from self diffusion
v v
For heat conduction, we set
1
dT
JQ  z    n v l c
3
dz
1
3
1
3
  nvlc  C vl
AcT
where c = heat capacity per particle
v  v = sound velocity
C = n c = heat capacity
Thermal Resistivity of Phonon Gas
Harmonic phonons: mfp l determined by collisions with boundaries & imperfections.
Anharmonic phonons: only U-processes contribute.
Gas:
Elastic collisions.
No T required.
κ=
Gas:
No net mass flow.
Inelastic collisions
with walls sets up T
& n gradients.
Finite κ.
Crystal:
N-processes only.
κ=.
Crystal:
U-processes.
Finite κ.
Umklapp Processes
2-D
square
lattice
Normal processes:
K1  K 2  K 3
Energy is
conserved:
Umklapp processes:
K1  K 2  K 3  G
1  2  3
Condition:
G0
1
K1 , K 2  G
2
T > θ: all modes excited → no distinction between N- & U- processes → l  1/T.
T < θ: probability of U- processes & hence l 1 exp(–θ /2T).
Imperfections
Low T → umklapp processes negligible.
Geometric effect dominates.
Size effect: l > D = smallest dimension of specimen.
K C v D T3
Dielectric crystals can have thermal
conductivities comparable to those of metals.
Sapphire (Al2O3): κ ~ 200W cm–1 K–1 at 30K.
Cu: max κ ~ 100W cm–1 K–1.
Metalic Ga: κ ~ 845W cm–1 K–1 at 1.8K.
( Electronic contributions dominate in metals. )
Highly purified c-NaF
Isotope effect on Ge.
Enriched: 96% Ge74.
Normal: 20% Ge70 , 8% Ge73 ,
37% Ge74 , 8% Ge76.